16 replies to this topic

[Prime]

You have a dozen (12) stones weighing a whole number of grams between 1 and 6 each. You can obtain one reference weight of your choosing.

What reference weight can you choose to be able to figure out the individual weights of the 12 stones using a balance device for any possibility that may exist therein?

 

For an encore: what is the maximum weight range of stones (1 to N) that you could solve using 2 reference weights of your choice? Provided you can have as many  stones as you need.

I don't believe, I have solved this one myself. We could make it a community project after the first question is answered.
 

HISTORICAL NOTE:

This problem originated on Brain Den. I constructed it based on Bonanova's problem Weighty  Thoughts: http://brainden.com/...4932--/?p=84107 few years ago.

Back then limited number of people participated. The solution found was for specific numbers in that problem (range 1 to 5) – not general. I'd like to give it another try.

[ThunderCloud]

I thought I had this one... but found a flaw in my reasoning. [solution withdrawn]

Edited by ThunderCloud, 23 February 2013 - 02:33 AM.

[CaptainEd]

First part: do we know that all weights between 1-6 are represented in the 12 stones? Or is it possible that we have, say, 12 stones each waiting 6?

[phil1882]

Spoiler for

seems to me you don't even need a reference weight. you can rank the weights using any standard sorting algorithm, and determine the mass by comparing multiple amounts of the same weight.

[ThunderCloud]

You have a dozen (12) stones weighing a whole number of grams between 1 and 6 each. You can obtain one reference weight of your choosing.

What reference weight can you choose to be able to figure out the individual weights of the 12 stones using a balance device for any possibility that may exist therein?

 

For an encore: what is the maximum weight range of stones (1 to N) that you could solve using 2 reference weights of your choice? Provided you can have as many  stones as you need.

I don't believe, I have solved this one myself. We could make it a community project after the first question is answered.
 

HISTORICAL NOTE:

This problem originated on Brain Den. I constructed it based on Bonanova's problem Weighty  Thoughts: http://brainden.com/...4932--/?p=84107 few years ago.

Back then limited number of people participated. The solution found was for specific numbers in that problem (range 1 to 5) – not general. I'd like to give it another try.

Spoiler for One method, though a bit of a cheat...

Using a reference weight of 5 grams, it would be easy to distinguish stones weighing 6 grams and 5 grams from the rest. Stones of lower weight could be sorted according to like weight, and then combined to see which combinations yield totals of 5 or 6+. The only difficulty is distinguishing a case where every stone weighs 3 grams from a case where every stone weighs 4 grams; either way, each is less than the reference weight, each is identical to every other, and any two are greater than the reference weight. The only method i can conceive to decide between these cases is by adding some other unknown but finely granulated weight to the scale (such as sand) to balance it... using this trick, one could determine which is the greater difference: one stone from the reference weight or the reference weight from two stones. If the weight difference between the reference weight and one stone is larger than between two stones and the reference, then the stones all weigh 3 grams; else they weigh 4.

[Prime]

First part: do we know that all weights between 1-6 are represented in the 12 stones? Or is it possible that we have, say, 12 stones each waiting 6?

There is no guaranty what weights are present/absent in your collection. The only guaranty is that each of the stones weighs a whole number of grams between 1 and 6.

12 stones each weighing 6 grams is one of the variations we must account for.

[CaptainEd]

I like T-cloud's cheat. It's not a reference weight, but just a found object that represents a difference. Good outside the box thinking!

[CaptainEd]

Part 2: you can have as many stones as you want. Which stones? Reference stones of my two chosen denominations? Or as many unknown stones?

[CaptainEd]

Part 2: you can have as many stones as you want. Which stones? Reference stones of my two chosen denominations? Or unknown stones?

[Prime]

Part 2: you can have as many stones as you want. Which stones? Reference stones of my two chosen denominations? Or as many unknown stones?

For the part 2, you can have 2 reference weights of your choice, as many stones as you want in the weight range from 1 to N. (Must find the largest N and the 2 reference weights.)

Let's solve the first part first. No tricks, no cheating, no different interpretations of the OP. If there is an ambiguity, I'll clarify it.